An airplane flies 2.00 × 102 km due west from city a to city b and then 3.00 × 102 km in the direction of 30.0° north of west from city b to city c. (a) in straight-line distance, how far is city c from city a? (b) relative to city a, in what direction is city c? (c) why is the answer only approximately correct

Explanation: See de image

Each parte correspond to a vector.

In principle let's look at the coordinates of each vector:

From A to B

x = -2.00 * 10 ^ 2km (west is negative)  and   y = 0km

From B to C

X(west)= -3.00*10^2*Cos30grades   and  y(North)= +3.00*10^2*Sin30grades

A straight-line from city A to C is the is the sum of the two vectors

x= -2.00 * 10 ^ 2km + -3.00*10^2*Cos30grades=  -459.81km

y=+3.00*10^2*Sin30grades=+150km

a) Distance= sqrt ((x^2)+(y^)2 = sqrt ( (-459.81km^2) +(+150km^2))= 483.66km

b) 18.07 grades Northwest

Tan(angle) = y/x

Angle= Tan^-1 (y/x)= Tan^-1 (150km/459.81km)=18.07 grades

c) Northwest